The smallest known moons in the solar system are very tiny compared to their planets.
For example, A moon one kilometer in diameter could orbit a gas giant whose rocky core - not counting the thick layers of atmospehre - is 10,000 kilometers in diameter and thus has a volume and mass 1,000,000,000,000 times as great as the moon.
In fact, if this list is sorted by radius, seven moons of Jupiter have radii of about 0.5 kilometer and thus diameters of about 1 kilometer, and two moons of Saturn have radii of about 0.15 and 0.33 kilometer and thus diameters of about 0.30 and 0.66 kilometer.
At the present time there are 202 known satellites of the four giant planets.
Some of them should have less than a trillionth of the mass of the rocky cores of the planets they orbit, let alone the total mass of their planets.
Jupiter has a mass of 318 Earth, Saturn has a mass of 95 Earths, Uranus has a mass of 14.5 Earths, and Neptune has a mass of 17 Earths. The approximate lower mass limit for a brown dwarf is about 13 times the mass of Jupiter and thus about 4,134 times the mass of Earth. But that is a very rough approximation.
Ganymede, most massive moon of Jupiter, has a mass of 0.0248 Earth, 0.0000779 that of Jupiter; Titan, most massive moon of Saturn, has a mass of 0.0225 Earth, 0.0002368 that of Saturn; Titania, most massive moon of Uranus, has a mass of 0.00059 Earth, 0.0000406 that of Uranus; and Triton, most massive moon of Neptune, has a mass of 0.003599 Earth, 0.0002177 that of Neptune.
Planetary astronomers divide terrestrial planets from gas planets by radius and not by mass, so a planet of the same mass could be either a terrestrial planet or a gas planet depending on how much athmospehre it has.
A gas dwarf is a gas planet with a rocky core that has accumulated a thick envelope of hydrogen, helium, and other volatiles, having as result a total radius between 1.7 and 3.9 Earth radii (1.7–3.9 R⊕). The term is used in a three-tier, metallicity-based classification regime for short-period exoplanets, which also includes the rocky, terrestrial-like planets with less than 1.7 R⊕ and planets greater than 3.9 R⊕, namely ice giants and gas giants.2
Theoretical studies of such planets are loosely based on knowledge about Uranus and Neptune. Without a thick atmosphere, it would be classified as an ocean planet instead.3 An estimated dividing line between a rocky planet and a gaseous planet is around 1.6-2.0 Earth radii. Planets with larger radii and measured masses are mostly low-density and require an extended atmosphere to simultaneously explain their masses and radii, and observations are showing that planets larger than approximately 1.6 Earth-radius (and more massive than approximately 6 Earth-masses) contain significant amounts of volatiles or H–He gas, likely acquired during formation.6 Such planets appear to have a diversity of compositions that is not well-explained by a single mass–radius relation as that found for denser, rocky planets. Similar results are confirmed by other studies. As for mass, the lower limit can vary widely for different planets depending on their compositions; the dividing mass can vary from as low as one to as high as 20 M⊕.
It seems that the least massive gas dwarf planets might possibly have masses similar to that of Earth.
So if a gaseous moon the mass of a gas dwarf planet would suffice, it is possible that a moon only as massive as Earth might possibly sometimes be gaseous.
If the limit between a gas giant planet and a brown dwarf is about 13 jupiter masses or about 4,134 Earth masses, if a gas planet at the edge with about 4,134 Earth masses had most massive moon with the Uranus-Titania mass ratio that moon would have a mass 0.167 Earth mass.
If a gas planet with a mass of 4,134 Earth masses has a largest moon with the Jupiter-Ganymede mass ratio, that moon would would have a mass of 0.322 Earth mass.
If a gas planet with a mass of 4,134 Earth masses has a largest moon with the Neptune-Triton mass ratio, that moon would would have a mass of 0.8999 Earth mass.
If a gas planet with a mass of 4,134 Earth masses has a largest moon with the Saturn-Titan mass ratio, that would would have a mass of 0.978 Earth mass.
So going by the examples in our solar system, it should be possible for a few of the very most massive giant planets on the border with brown dwarfs to have moons as massive as the planet Earth, and it is possible that objects no more massive than Earth sometimes becone gas dwarfs.
But consider the example of the Earth and the Moon. The moon has mass of 0.0123 Earth mass. If a giant planet on the border with brown dwarfs with a mass of 4,134 Earths had a moon with that mass ratio that moon would have a mass of 50.848 Earths.
Of course it is believed that the moon formed from a debris ring after another planet collided with Earth, and such an origin for a moon of a giant planet might leave that moon without any gas.
Pluto has a mass of 0.0022 Earth, and its largest moon Charon has a mass of 0.00025 Earth, a mass ratio of about 0.1136. If a giant planet on the border with brown dwarfs with a mass of 4,134 Earths had a moon with that mass ratio that moon would have a mass of 469.772 Earth.
Of course Pluto and Charon are also believed to have formed from an impact.
At the preent time exmoons are near the limits of detectability and no proposed exomoons have been confirmed.
Most of the exomoon candidtates so far are probably more massive that Earth, and so might potentially count as superearths, gas dwarfs, ice giants, or gas giants.
And if there are ever confirmed moons of Earth mass or greater orbiting objects confirmed to be planets instead of brown dwarfs, the planet to moon mass ratios in our solar system will be shown to be medium values insted of extreme values.
I note that Titan, the largest moon of Saturn, has a mass of 0.0225 Earth, between Callisto and Ganymede, the largest moons of Jupiter, with masses of 0.018 and 0.248 Earth respectively.
Their escape velocitiea are Callisto 2.440 kilometers per second, Titan 2.639 kilometers per second, and Ganymede 2.741 kilometers per second.
Since Titan is farther from the Sun and colder, it should have a slightly better ability to retain an atmosphere than Callisto and Ganymede do.
But the atmosphere of Titan is literally billions of times as dense and has literally bilions of times the total mass as the atmospheres of Callisto and Ganymede. And the reasons for that difference are not known as far as I know.
So if two exomoons with identical mass orbit giant exoplanets and recieve the same amounts of radiation from their stars, one could be a basically airless rock and another could have an atmosphere many times as dense at Earth's and perhaps count as a gas dwarf, depending on varius factors.